Abandoned Hall

No. Abandoned Hall's ability triggers "when a player plays a spell," and a countered spell is not considered played.

A card is only played once the entire play process has finished in its entirety.^[350.1] Abilities that trigger on playing a card fire when that act has been completed by the card's resolution;^[419.4.a] if a game effect prevents resolution — for example because the spell was countered — the card is not considered to have been played and no abilities that trigger on playing cards fire.^[419.4.a.1] This follows directly from what countering does: a countered card is cleared from the chain and is not considered to have been played.^[425.1.b]

Only if it is at Abandoned Hall after the spell fully resolves.

A card is only considered "played" once the entire play process has finished,^[350.1] which for spells means after all instructions have been executed and the card is placed in the trash. Abandoned Hall's triggered ability therefore fires after the spell resolves, not when it is placed on the chain.^{[419.4.a][383.2.c]} By then, a unit moved into Abandoned Hall is present and can be chosen, while a unit moved away no longer qualifies as "a unit they control here."

Yes. Spells like Arcane Shift and Thrill of the Hunt add the unit to the chain as a pending item during their own resolution, pausing its play process while the spell continues.^{[354.2][354.3]} When the spell finishes resolving, it is fully played and Abandoned Hall's triggered ability becomes pending after the unit.^{[350.1][419.4.a][383.2.c]} Because pending items are finalized in the order they were appended,^[337.1.b] the unit is finalized and placed on the board first — resolving immediately on finalization^[337.2] — before the triggered ability's target is chosen. By that point the unit is already present at Abandoned Hall and is a valid choice for the buff.

Taking Arcane Shift as an example: